Homotopy colimits and global observables in Abelian gauge the

26 Pages·2017·0.41 MB·English

by  Christine Middleton

Homotopy colimits and global observables
in Abelian gauge theory
Marco Beninia, Alexander Schenkelb and Richard J. Szaboc
Department of Mathematics, Heriot-Watt University,
Colin Maclaurin Building, Riccarton, Edinburgh EH14 4AS, United Kingdom.
5
Maxwell Institute for Mathematical Sciences, Edinburgh, United Kingdom.
1
0 The Tait Institute, Edinburgh, United Kingdom.
2

y Email: a [email protected] , b [email protected] , c [email protected]
a
M
May 2015

9
2
Abstract
]
h
We study chain complexes of field configurations and observables for Abelian gauge
p
theory on contractible manifolds, and show that they can be extended to non-contractible
-
h manifolds by using techniques from homotopy theory. The extension prescription yields
t
a functors from a category of manifolds to suitable categories of chain complexes. The ex-
m tended functors properly describe the global field and observable content of Abelian gauge
[ theory, while the original gauge field configurations and observables on contractible mani-
folds are recoveredup to a natural weak equivalence.
2
v
9 Report no.: EMPG–15–04
3
8
8 Keywords: Abelian gauge theory, global configurations and observables, chain complexes,
0
homotopy limits and colimits
.
3
0
5 MSC 2010: 70S15, 81T13, 55U15, 55N20
1
:
v
i
X Contents
r
a
1 Introduction and summary 2
2 Local gauge field configurations and observables 5
2.1 Groupoids and cosimplicial algebras . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Abelian gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Homotopy limits and global gauge field configurations 9
3.1 Canonical diagrams of gauge field configurations . . . . . . . . . . . . . . . . . 9
3.2 Homotopy limit of canonical diagrams . . . . . . . . . . . . . . . . . . . . . . . 10
3.3 Deligne complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.4 Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.5 Functor extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1
4 Homotopy colimits and global gauge field observables 14
4.1 Canonical diagrams of gauge field observables . . . . . . . . . . . . . . . . . . . 14
4.2 Homotopy colimit of canonical diagrams . . . . . . . . . . . . . . . . . . . . . . 14
4.3 Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.4 Functor extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.5 Natural pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
A Dold-Kan correspondence and Moore complex 18
B Homotopy limits and colimits for chain complexes 19
B.1 Homotopy limits for non-negatively graded chain complexes . . . . . . . . . . . 19
B.2 Homotopy colimits for non-positively graded chain complexes . . . . . . . . . . 21
1 Introduction and summary
In a classical field theory (without gauge symmetry) the field configurations on a manifold M
are described in terms of the set of sections F(M) := Γ∞(E ) of some natural fibre bundle
M
E over M. For example, for a real scalar field theory the field configurations on M are given
M
by the set C∞(M) of functions on M, which is the set of sections of the trivial line bundle
R×M → M. Naturality allows us to regard F : Manop → Sets as a functor from the category
of (say, oriented m-dimensional) manifolds to the category of sets. The sheaf property of the
set of sections of a fibre bundle allows us to capture all information about the global field
configurations F(M) on a manifold M in terms of the local field configurations in an open
cover {U : i ∈ I} of M. More precisely, the sheaf property says that the set F(M) can be
i
recovered (up to isomorphism) from any open cover {U : i∈ I} of M by taking the limit
i
F(M) ∼= // lim F(Ui) →→ F(Uij) (1.1)
(cid:16)Qi Qi,j (cid:17)
in the category Sets. Here denotes the categorical product in Sets and as usual we denote
the intersections by U := UQ∩U .
ij i j
Another essential ingredient of a classical field theory is the characterization of the ob-
servables of the theory, which is usually done by specifying for each manifold M a suitable
algebra A(M) of functions on F(M).1 Following [BFV03], an important guiding principle for
the choice of the observable algebras A(M) is the requirement of functoriality of the assign-
ment A : Man → Alg, where Alg is a suitable category of algebras whose details depend on
the context. Another reasonable requirement is the cosheaf property of A, which would allow
us to capture all information about the global observables A(M) on a manifold M in terms of
the local observables in an open cover {U : i ∈ I} of M. More precisely, the cosheaf property
i
demands that the algebra A(M) can be recovered (up to isomorphism) from any open cover
{U : i∈ I} of M by taking the colimit
i
A(M) oo ∼= colim A(Uij) →→ A(Ui) (1.2)
(cid:16)`i,j `i (cid:17)
in the category Alg. Here denotes the categorical coproduct in Alg.
`
1 The algebras A(M) typically carry a Poisson structure for classical field theories defined on globally hy-
perbolic Lorentzian manifolds, see e.g. [BFR12], or they are noncommutative algebras after quantization, see
e.g. [BFV03]. These additional structures will not play a role in the present paper because we do not consider
dynamical aspects of field theories or quantization.
2
Whenstudyingexplicitexamples ofclassical (andalsoquantum)fieldtheories itmightvery
well happen that one can construct rather easily the field configurations F(M) and the observ-
ables A(M) of the theory on a special class of manifolds, e.g. on the category of contractible
manifolds Man , but that the construction becomes much more involved for non-contractible
(cid:13)c
manifolds. Reasons for this might be global features, such as non-trivial bundles and topo-
logical charges, which are related to topologically non-trivial manifolds. In such a situation
one would obtain two functors F : Manop → Sets and A : Man → Alg describing the field
(cid:13)c (cid:13)c
configurations and observables of the theory only on contractible manifolds, and the goal is to
then ‘extend’ these functors to the category of all manifolds Man. In view of the desired sheaf
and cosheaf properties, a reasonable procedure for obtaining such extensions is to define the
field configurations F(M) and the observables A(M) on a generic manifold M in terms of the
limit or, respectively, the colimit of a diagram induced by a suitable open contractible cover of
M. In the context of algebraic quantum field theory, such a procedure has been proposed by
Fredenhagen and it is called the ‘universal algebra’, see e.g. [Fre90, Fre93, FRS92].
In gauge theories the structures discussed above become considerably more complicated.
First of all, the gauge field configurations on a manifold M are not described by a set, but by
a groupoid. For example, the field configurations of gauge theory with structure group G on a
manifold M are described by the groupoid with objects given by all principal G-bundles over
M endowed with a connection and morphisms given by all principal G-bundle isomorphisms
compatible with the connections (i.e. gauge transformations). Instead of forming a sheaf, the
collection of these groupoids for all manifolds M forms a stack, see e.g. [Fan01, Vis05] for
an introduction. For our purposes, a more explicit and also more suitable characterization of
stacks in terms of homotopy sheaves of groupoids has been developed by Hollander [Hol08a].
A stack is the same thing as a functor G : Manop → Groupoids to the category of groupoids
which satisfies the homotopy sheaf property, i.e. for any manifold M the groupoid G(M) can
be recovered (up to weak equivalence) from any open cover {U : i ∈ I} of M by taking the
i
homotopy limit
G(M) ∼ //holim G(Ui)→→ G(Uij)→→→ G(Uijk) →→→→ ··· (1.3)
(cid:16)Qi Qi,j iQ,j,k (cid:17)
in the category Groupoids.2 Note that forming the coarse moduli spaces (i.e. the gauge orbit
spaces) of a homotopy sheaf in general does not yield a sheaf. Hence the groupoid point of
view is essential for ‘gluing’ local gauge field configurations to global ones.
Classical observables for gauge theories may be described by taking suitable ‘function al-
gebras’ on groupoids, which can be modeled by cosimplicial algebras or differential graded
algebras, see Section 2 below for details. A natural requirement for the choice of such ‘func-
tion algebras’ is again functoriality, i.e. we seek a functor B : Man → cAlg to the category
of cosimplicial algebras (or a functor B : Man → dgAlg to the category of differential graded
algebras). Instead of the cosheaf property which appears in ordinary field theories, this func-
tor should satisfy the homotopy cosheaf property, i.e. the cosimplicial (or differential graded)
algebra B(M) can be recovered (up to weak equivalence) from any open cover {U : i ∈ I} of
i
M by taking the homotopy colimit
B(M) oo ∼ hocolim ··· →→→→ B(Uijk) →→→ B(Uij) →→ B(Ui) (1.4)
(cid:16) i`,j,k `i,j `i (cid:17)
in the category cAlg (or in the category dgAlg).3
2For a concise and very readable introduction to homotopy theory and model categories see [DS95].
3See [CC04, Jar97] for details on the relevant model category structureson cAlg and dgAlg.
3
In gauge theories we are exactly in the situation where the groupoids of field configurations
are rather simple and explicit for contractible manifolds M, while they are much harder to de-
scribefornon-contractiblemanifolds. ThisisbecauseonacontractiblemanifoldM allprincipal
G-bundles are isomorphic to the trivial principal G-bundle M ×G→ M. Consequently, gauge
field observables on contractible manifolds are also much easier to describe than those on non-
contractible manifolds. Hence we can rather easily get two functors G : Manop → Groupoids
(cid:13)c
and B : Man → cAlg (or B : Man → dgAlg) which describe gauge field configurations
(cid:13)c (cid:13)c
and observables only on contractible manifolds. The goal is then to extend these two func-
tors to the category Man by taking homotopy limits or, respectively, homotopy colimits of
diagrams induced by suitable open contractible covers of manifolds M. From the perspec-
tive of algebraic quantum field theory, these constructions may be interpreted as a gauge
theoretic (or homotopy theoretic) version of Fredenhagen’s ‘universal algebra’ construction.
Let us emphasize again the importance of describing gauge field configurations in terms of
groupoids and observables in terms of cosimplicial (or differential graded) algebras, instead
of working with gauge orbit spaces and gauge invariant observable algebras. As an explicit
example of what may go wrong when not doing so, see [DL12, FL14] where the ‘universal
algebra’ has been constructed for gauge invariant observable algebras of Abelian gauge the-
ory. These ‘universal algebras’ fail to produce the correct global gauge invariant observable
algebras [BSS14] because they neglect flat connections and violate the quantization condition
for magnetic charges in the integral cohomology H2(M,Z). See also [BDHS14, BDS14] for a
presentation of the global gauge invariant observable algebras for a fixed but arbitrary prin-
cipal bundle and [FP03, DS13, SDH14, CRV13, Ben14] for the trivial principal bundle. Cer-
tain aspects of non-Abelian gauge theories and also gravity in this context have been studied
in [Hol08b, FR13, CRV12, BFR13, Kha14, Kha15].
In this paper we shall make explicit and test the above ideas for constructing global gauge
field configurations and observables by homotopy theoretic techniques. We shall consider the
simplest example of a classical gauge theory, namely that whose structure group is the circle
group G = T = U(1). From the perspective of differential cohomology, there already exist
several models for the groupoid of gauge potentials in this case which have been discussed from
theperspectiveof‘locality’ of (generalized) Abelian gaugetheories: Thecategory ofdifferential
cocycles constructed by [HS05] is based on singular cochains (see also [Sza12, Section 2.4]),
while the Cˇech theoretic model of [FW99] is somewhat closer in spirit to our approach (see
also [Fre00, Example 1.11]); see also [BM06] for a more heuristic proposal. A similar functorial
description of abelian gauge theories is discussed by [FSS15]. To simplify our constructions, we
shall study this gauge theory on a purely kinematical level, leaving out both dynamical aspects
(i.e. Maxwell’s equations) and quantization for the moment.
Theoutline of the remainder of this paperis as follows: In Section 2 we give an explicit and
very useful description of the groupoids of gauge field configurations on contractible manifolds
in terms of chain complexes of Abelian groups. This formulation allows us to identify a simple
class of gauge field observables, given by smooth group characters, which also forms a chain
complex of Abelian groups. Our constructions are functorial in the sense that we obtain
a functor C : Manop → Ch (Ab) describing gauge field configurations and a functor O :
(cid:13)c ≥0
Man → Ch (Ab) describing observables on the category of contractible manifolds Man .
(cid:13)c ≤0 (cid:13)c
In Section 3 we shall construct an extension of the functor C : Manop → Ch (Ab) to the
(cid:13)c ≥0
category Man of all manifolds by using homotopy limits. For this we first show that any
manifold M has a canonical open cover by contractible subsets, which induces a canonical
diagram of gauge field configurations on M. We compute explicitly the homotopy limit of
this diagram and show that it is isomorphic to the Deligne complex in the canonical cover.
This will imply that our homotopy limit describes all possible gauge field configurations on M,
including also non-trivial principal T-bundles whenever H2(M,Z) 6= 0. As the canonical cover
4
isfunctorial, itiseasy toprovethattheglobalfieldconfigurationsgiven bythehomotopy limits
are described by a functor Cext : Manop → Ch (Ab). We shall show that this functor is an
≥0
extension (up to a natural quasi-isomorphism) of our original functor C : Manop → Ch (Ab).
(cid:13)c ≥0
In Section 4 we shall focus on the gauge field observables and construct an extension of the
functor O : Man → Ch (Ab) to the category Man by using homotopy colimits. Similarly
(cid:13)c ≤0
to the gauge field configurations, we obtain a canonical diagram of gauge field observables
on any manifold M and we compute explicitly the homotopy colimits. Functoriality of the
global observables Oext : Man → Ch (Ab) is again a simple consequence of functoriality of
≤0
the canonical cover. We then show that this functor is an extension (up to a natural quasi-
isomorphism) of our original functor O : Man → Ch (Ab). Finally, we construct a natural
(cid:13)c ≤0
pairingbetween global gauge field configurations and observables, which allows us to show that
our class of observables separates all possible gauge field configurations. Two appendices at
the end of the paper summarize some of the more technical details which are used in the main
text. In Appendix A we review the (dual) Dold-Kan correspondence, which is an important
technical tool for our constructions. In Appendix B we summarize the explicit techniques to
compute homotopy (co)limits for chain complexes of Abelian groups given in [Dug, Section
16.8] and [Rod14].
2 Local gauge field configurations and observables
In this section we consider gauge fields on contractible manifolds; in this paper all manifolds
considered are oriented.
2.1 Groupoids and cosimplicial algebras
Let G be a (matrix) Lie group, g its Lie algebra and M a contractible manifold. Then all
principalG-bundlesover M areisomorphicto the trivial G-bundle, andthefield configurations
ofgaugetheorywithstructuregroupGonM aredescribedbytheg-valuedone-formsΩ1(M,g);
elements A∈ Ω1(M,g) are typically called ‘gauge potentials’. Recall that gauge theory comes
withadistinguishednotionofgaugegroup,thegroupofverticalautomorphismsoftheprincipal
G-bundle. In the present case the gauge group may be identified with the group of G-valued
smooth functions C∞(M,G) and it acts on gauge potentials from the left via
ρ: C∞(M,G)×Ω1(M,g) −→ Ω1(M,g) , (g,A) 7−→ ρ(g,A) = gAg−1+gdg−1 , (2.1)
where d denotes the exterior derivative.
Havingavailablebothgaugepotentialsandgaugetransformations,onecanaskwhichmath-
ematical structure is suitable for describing the relevant field content of gauge theory on M.
The most obvious option is to take the orbit space Ω1(M,g)/C∞(M,G) under the ρ-action,
which identifies all gauge potentials that differ by a gauge transformation; this is often called
the ‘gauge orbit space’. However, there are several problems with the orbit space construction:
First, even though both Ω1(M,g) and C∞(M,G) can be equipped with a suitable (locally
convex infinite-dimensional) smooth manifold structure, the orbit space is in general singu-
lar [ACMM86, ACM89]. Second, forming the orbit space inevitably leads to a substantial loss
of information; even though we can still decide whether or not two gauge potentials A and
A′ are gauge equivalent, we cannot keep track of the gauge transformation g that identifies
A with A′ when they are equivalent. The latter information is essential whenever one wants
to obtain global field configurations of gauge theory on a topologically non-trivial manifold M
by ‘gluing’ local configurations in contractible patches. A classic example is Dirac’s famous
magnetic monopole which represents the Chern class in Abelian gauge theory with structure
group the circle group G= T = U(1): Its construction is based on gauge potentials A and A
1 2
5
on an open cover {U ,U } of a topologically non-trivial manifold M subject to the requirement
1 2
A | −A | = g dg−1 for some fixed g ∈ C∞(U ,T) on the overlap U = U ∩U . This
2 U12 1 U12 12 12 12 12 12 1 2
operation of ‘gluing up to gauge transformations’ cannot be described in terms of gauge orbit
spaces.
Inordertosolvetheseandotherproblems,amoremodernperspectivesuggeststhat,instead
of looking at gauge orbits, one should organize the gauge potentials and gauge transformations
into a groupoid. Recall that a groupoid is a small category in which every morphism is invert-
ible. The groupoid corresponding to gauge theory with structure group G on a contractible
manifold M is simply the action groupoid C∞(M,G)⋉Ω1(M,g) ⇒ Ω1(M,g): The set of ob-
jects is Ω1(M,g) and the set of morphismsis C∞(M,G)×Ω1(M,g), an element (g,A) of which
should be interpreted as an arrow
(g,A)
ρ(g,A) ←−−−− A (2.2)
starting at A and ending at the gauge transform (2.1) of A by g. Two morphisms (g′,A′) and
(g,A) are composable whenever A′ = ρ(g,A) and the composition reads as (g′,A′)◦(g,A) =
(g′g,A). Theidentitymorphismsareid = (e,A),whereeistheidentityelementinC∞(M,G),
A
i.e. the constant function to the identity element of the structure group G. As an aside, note
that one can usethe techniques of [ACMM86] to realize that this action groupoid is moreover a
(locally convex infinite-dimensional) Lie groupoid, i.e. a groupoid carrying a smooth structure.
This smooth structure will not play a role in the present paper, since we will shortly restrict
ourselves to Abelian gauge theory with structure group G= T, which can be studied in purely
algebraic terms. However, in studies of non-Abelian gauge theory the smooth structures will
play an important role.
Instead of using groupoids, we may equivalently organize the gauge potentials and gauge
transformations into a simplicial set (or even a simplicial manifold if we use the smooth struc-
ture discussed above). Recall that a simplicial set is a collection {Sn}n∈N0 of sets together with
face maps ∂n :S → S , for n ≥ 1 and i= 0,1,...,n, and degeneracy maps ǫn :S → S ,
i n n−1 i n n+1
for n ≥ 0 and i = 0,1,...,n, satisfying simplicial identities, see e.g. [GJ99, Section I.1]. The
simplicial set corresponding to our groupoid (which is called its nerve) may be depicted as
Ω1(M,g) oooo C∞(M,G)×Ω1(M,g) oooooo C∞(M,G)×2 ×Ω1(M,g) oooooooo ··· , (2.3)
wherethearrowsarethefacemapsandwehavesuppressedthedegeneracy mapsfornotational
convenience. Explicitly, the face maps read as
∂n : C∞(M,G)×n ×Ω1(M,g) −→ C∞(M,G)×n−1 ×Ω1(M,g) ,
i
(g ,...,g ,A) , for i = 0 ,
2 n

(g ,...,g ,A) 7−→ (g ,...,g g ,...g ,A) , for i = 1,...,n−1 ,
1 n  1 i i+1 n

(g ,...,g ,ρ(g ,A)) , for i = n ,
1 n−1 n

 (2.4a)
and the degeneracy maps read as
ǫn :C∞(M,G)×n ×Ω1(M,g) −→ C∞(M,G)×n+1 ×Ω1(M,g) ,
i
(g ,...,g ,A) 7−→ (g ,...,g ,e,g ,...,g ,A) . (2.4b)
1 n 1 i i+1 n
The simplicial set perspective has the advantage of making clear how to describe gauge
theory observables. Interpreting (2.3) as the simplicial set of gauge field configurations, it is
natural to model (classical) observables as functions on it. This can be done by taking the
algebra of complex-valued functions C(−,C) on each degree of (2.3). Doing so, a cosimplicial
6
algebra is obtained by dualizing the face and degeneracy maps to co-face and co-degeneracy
maps under the contravariant functor C(−,C) : Sets → Alg between sets and algebras. Re-
stricting to infinitesimal gauge transformations, this picture reduces nicely to the well-known
BRST formalism, see [FR12] for a presentation of this topic in the context of the algebraic ap-
proach to field theory. By the dual Dold-Kan correspondence (see Appendix A) we can regard
our cosimplicial algebra as a differential graded algebra (dg-algebra), see also [CC04] for more
details. This dg-algebra can be ‘linearized’ via a procedurecalled the van-Est map (here we re-
quire the smooth structure mentioned above) to yield the BRST algebra (Chevalley-Eilenberg
dg-algebra) of gauge theory, see e.g. [Cra03]. It is important to stress that this linearization
procedure neglects finite gauge transformations and hence leads to an incomplete description
of gauge theory. Our cosimplicial algebra (or its associated dg-algebra) should be interpreted
as an improvement of the usual BRST algebra, which keeps track of all gauge transformations
and not only of the infinitesimal ones; in fact, finite gauge transformations are essential for
gluing local field configurations to global ones. Using the analogy with the BRST formalism,
we may call the factors C∞(M,G) in (2.3) the ‘ghost fields’. Notice that these ghost fields are
non-linear in the sense that they are functions with values in the structure group, while the
ordinary ghost fields in the BRST formalism are described by the tangent space C∞(M,g) at
the identity e∈ C∞(M,G) and hence they are linear.
2.2 Abelian gauge theory
In the remainder of this paper we shall fix the structure group G = T with Lie algebra t= iR
and hence consider only Abelian gauge theory. Then the structures described above simplify
considerably. In particular, all sets appearing in (2.3) naturally become Abelian groups with
respect to the direct product group structure given by
(g ,...,g ,A)(g′,...,g′ ,A′) := (g g′,...,g g′,A+A′) . (2.5)
1 n 1 n 1 1 n n
Moreover, the action of the gauge group on gauge potentials (2.1) simplifies to ρ(g,A) =
A+gdg−1, and it is easy to show that the face and degeneracy maps (2.4) are group homo-
morphisms. It follows that the simplicial set (2.3) is a simplicial Abelian group, which under
the Dold-Kan correspondence can be identified with a non-negatively graded chain complex of
Abelian groups, see Appendix A. This chain complex is called the normalized Moore complex
and in the present case it reads explicitly as
C(M):= C(M) , δ := Ω1(M,t)⊕C∞(M,T), δ , (2.6)
n
(cid:16)M (cid:17) (cid:16) (cid:17)
n≥0
whereΩ1(M,t)sitsindegree0andC∞(M,T)sitsindegree1. Asaconvenientsignconvention,
we shall take as the differential (of degree −1) in C(M) the negative of the differential in the
normalized Moore complex (A.2), i.e.
δ(A⊕g) = (gdg−1)⊕0 . (2.7)
M being contractible, the homology H of the chain complex C(M) is given by
∗
Ω1(M,t)
H (C(M)) = , H (C(M)) ≃ T , (2.8)
0 dC∞(M,t) 1
which gives the gauge orbit space in degree 0 and the global constant gauge transformations
in degree 1. Notice that the first homology group is not a vector space, but only an Abelian
group. This feature naturally distinguishes between the Abelian gauge theories with structure
groups G = T and G = R: Both theories have the same zeroth homology (i.e. the same gauge
7
orbit space) on contractible manifolds, but they differ in the first homology which is always
isomorphic to G.
For Abelian gauge theory we also obtain a distinguished class of observables: Since in this
case (2.3) is a simplicial Abelian group, instead of all complex-valued functions C(−,C) on
each degree, we can take only those functions which are group characters, i.e. homomorphisms
of Abelian groups HomAb(−,T) to the circle group. The group characters do not form an
algebra, but rather an Abelian group called the character group; of course one can generate an
algebra by the group characters, but this will not be done in the present paper. The character
group should be interpreted as a generalization of the vector space of linear observables for a
real scalar field theory, which also does not form an algebra, but which generates a polynomial
algebra. Taking the character groups HomAb(−,T) in each degree of (2.3) gives rise to a
cosimplicial Abelian group because all face and degeneracy maps dualize to co-face and co-
degeneracy maps. Under the dual Dold-Kan correspondence this can be identified with a
non-positively graded chain complex of Abelian groups, see Appendix A, which for our model
explicitly reads as
C∞(M,T)∗⊕Ω1(M,t)∗, δ∗ , (2.9)
(cid:16) (cid:17)
where −∗ := HomAb(−,T). Here C∞(M,T)∗ sits in degree −1 and Ω1(M,t)∗ sits in degree 0,
while the differential δ∗ (of degree −1) is the dual of the differential (2.7). Using the smooth
character groups as in [BSS14], the chain complex (2.9) can be restricted to
O(M) := O(M) , δ∗ := Ωm (M)⊕Ωm−1(M), δ∗ , (2.10)
n c,Z c
(cid:16)M (cid:17) (cid:16) (cid:17)
n≤0
where m is the dimension of M and the subscript indicates differential forms of compact
c
support. By Ωm (M) we denote the subgroup of Ωm(M) consisting of compactly supported
c,Z c
top-degree forms which integrate to an integer, i.e. χ ∈ Ωm (M) if and only if χ ∈ Z. It is
c,Z M
instructive to explain in more detail how (2.10) defines group characters on (2R.6) and to give
an explicit formula for δ∗: Let us define the non-degenerate pairing
−,− : O(M)×C(M) −→ T ,
M
(cid:10) (cid:11)
(χ⊕ϕ,A⊕g) 7−→ exp A∧ϕ+log(g)χ . (2.11)
(cid:16)ZM (cid:0) (cid:1)(cid:17)
We observe that (2.11) is a bi-character, i.e.
χ⊕ϕ+χ′⊕ϕ′,A⊕g = χ⊕ϕ,A⊕g χ′⊕ϕ′,A⊕g , (2.12a)
M M M
(cid:10) (cid:11) (cid:10) (cid:11) (cid:10) (cid:11)
χ⊕ϕ,A⊕g+A′⊕g′ = χ⊕ϕ,A⊕g χ⊕ϕ,A′ ⊕g′ , (2.12b)
M M M
(cid:10) (cid:11) (cid:10) (cid:11) (cid:10) (cid:11)
and that it is compatible with the gradings of O(M) and C(M) if we take the target T to sit in
degree 0. The differential δ∗ in O(M) is defined via the duality induced by (2.11); we compute
χ⊕ϕ,δ(A⊕g) = χ⊕ϕ,(gdg−1)⊕0)
M M
(cid:10) (cid:11) (cid:10) (cid:11)
= exp − dlog(g)∧ϕ
(cid:16) ZM (cid:17)
= dϕ⊕0,A⊕g =: δ∗(χ⊕ϕ),A⊕g , (2.13)
M M
(cid:10) (cid:11) (cid:10) (cid:11)
from which we find δ∗(χ⊕ϕ) = dϕ⊕0. Recalling that in the present section all manifolds are
contractible, it follows that the homology H of the chain complex (2.10) is given by
∗
H (O(M)) ≃ Z , H (O(M)) =Ωm−1(M) := Ker d: Ωm−1(M) → Ωm(M) . (2.14)
−1 0 c,d c c
(cid:0) (cid:1)
8
Comparingthesegroupswith(2.8), weseethatH (O(M)) contains exactly thegroupcharac-
−1
ters on H (C(M)) and that H (O(M)) is given by the (smooth) group characters on the gauge
1 0
orbit space H (C(M)), i.e. the zeroth homology of O(M) describes gauge invariant group char-
0
acters and hence gauge invariant observables of that kind.
All of these constructions are functorial. Let us denote by Man the following category
(cid:13)c
of contractible manifolds: The objects in Man are all contractible oriented manifolds of
(cid:13)c
a fixed dimension m (which we suppress from the notation) and the morphisms in Man
(cid:13)c
are all orientation preserving open embeddings. The chain complexes in (2.6) are described
by a functor C : Manop → Ch (Ab), where Manop denotes the opposite category (i.e. the
(cid:13)c ≥0 (cid:13)c
category withreversed arrows)of Man andCh (Ab)isthecategory ofnon-negatively graded
(cid:13)c ≥0
chain complexes of Abelian groups; the functor C assigns to any object M in Man the chain
(cid:13)c
complex C(M) given in (2.6) and to any morphism fop :M′ → M in Manop (i.e. any morphism
(cid:13)c
f :M → M′ in Man ) the morphism of chain complexes
(cid:13)c
C(fop):= f∗ : C(M′) −→ C(M) , A′⊕g′ 7−→ f∗(A′)⊕f∗(g′) , (2.15)
wheref∗ isthepull-back offunctions/differentialformsalongf. Similarly, thechaincomplexes
in (2.10) are described by a functor O : Man → Ch (Ab) to the category of non-positively
(cid:13)c ≤0
graded chain complexes of Abelian groups; the functor O assigns to any object M in Man
(cid:13)c
the chain complex O(M) given in (2.10) and to any morphism f : M → M′ in Man the
(cid:13)c
morphism of chain complexes
O(f) := f : O(M) −→ O(M′) , χ⊕ϕ 7−→ f (χ)⊕f (ϕ) , (2.16)
∗ ∗ ∗
where f is the push-forward of compactly supported differential forms along f. It follows that
∗
the pairing (2.11) is natural in the sense that the diagram
id×f∗
O(M)×C(M′) //O(M)×C(M) (2.17)
f∗×id h−,−iM
(cid:15)(cid:15) (cid:15)(cid:15)
O(M′)×C(M′) // T
h−,−iM′
commutes for all morphisms f : M → M′ in Man .
(cid:13)c
3 Homotopy limits and global gauge field configurations
Our functor C : Manop → Ch (Ab) given by (2.6) and (2.15) describes the chain complexes
(cid:13)c ≥0
of gauge field configurations (together with gauge transformations) of Abelian gauge theory
on contractible manifolds. Notice that for an arbitrary manifold M the chain complex in (2.6)
does not necessarily describe all gauge field configurations on M: For example, M might have
a non-trivial second cohomology H2(M,Z) 6= 0, hence allowing for non-trivial principal T-
bundles which are not captured by (2.6). The goal of this section is to extend the functor
C : Manop → Ch (Ab) to the category Man of all oriented m-dimensional manifolds (with
(cid:13)c ≥0
morphisms given by orientation preserving open embeddings) by using homotopy theoretic
techniques.
3.1 Canonical diagrams of gauge field configurations
To any object M in Man we can assign the category D(M) of all contractible open subsets
U ⊆ M with morphisms given by subset inclusions U ⊆ V. The set of objects D(M) of
0
D(M) is an open cover of M, i.e. U = M. Notice that, even though every U ∈
U∈D(M)0
S
9
D(M) is contractible, the open cover D(M) is not a good cover: The intersection of two
0 0
contractible open subsets fails to be contractible in general. This is not problematic since our
main constructions do not use intersections. To any morphism f : M → M′ in Man we can
assign the functor D(f): D(M)→ D(M′) which maps any contractible open subset U ⊆ M to
its image f(U)⊆ M′. In summary, we have defined a functor
D :Man −→ Cat (3.1)
to the category Cat of small categories.
Let now M be any object in Man. Notice that any object U in D(M) carries a canonical
orientation by pulling back the orientation of M under the subset inclusion U ⊆ M. Hence
we may regard D(M) as a subcategory of Man and we can restrict the functor C : Manop →
(cid:13)c (cid:13)c
Ch (Ab) to a functor on D(M)op, which we shall denote by C : D(M)op → Ch (Ab). The
≥0 M ≥0
functor C assigns to any contractible open subset U ⊆ M the chain complex C(U) given by
M
(2.6) and to any subset inclusion U ⊆ V the restriction map | :C(V) → C(U) given by (2.15)
U
for the subset inclusion U ⊆ V. Given now any morphism f : M → M′ in Man, the functor
(3.1) gives a functor D(f) : D(M) → D(M′), which defines a functor (denoted by the same
symbol) D(f) : D(M)op → D(M′)op between the opposite categories. Hence we obtain two
functors
C :D(M)op −→ Ch (Ab) , C ◦D(f): D(M)op −→ Ch (Ab) , (3.2)
M ≥0 M′ ≥0
with the same source and target category. The pull-back (2.15) then defines a natural trans-
formation (denoted by the same symbol)
f∗ : C ◦D(f)=⇒ C . (3.3)
M′ M
3.2 Homotopy limit of canonical diagrams
In this subsection we shall fix an arbitrary object M in Man and study the homotopy limit of
the canonical diagram C : D(M)op → Ch (Ab) given in (3.2), which we denote by
M ≥0
Cext(M) := Cext(M) , δ := holim(C ) . (3.4)
n M
(cid:16)M (cid:17)
n≥0
We use the techniques of [Dug, Section 16.8] and [Rod14], which we have summarized in detail
in Appendix B.1. Recall the explicit form of the functor C :Manop → Ch (Ab) given by (2.6)
(cid:13)c ≥0
and (2.15). In order to compute the homotopy limit (3.4) of its restriction C : D(M)op →
M
Ch (Ab), we first take the cosimplicial replacement of this functor. This yields a cosimplicial
≥0
objectinCh (Ab),see(B.1),(B.2)and(B.3)fordetailedexpressions. Usingtheco-normalized
≥0
Moore complex (A.3), weassign to thiscosimplicial object inCh (Ab)adoublechain complex
≥0
inCh (Ch (Ab)),cf.(B.5),whichforourpresentfunctorC : D(M)op → Ch (Ab)explicitly
≤0 ≥0 M ≥0
reads as
Ω1(U,t) oo δh C∞(U,T) (3.5)
QU QU
δv δv
(cid:15)(cid:15) (cid:15)(cid:15)
Ω1(U,t) oo δh C∞(U,T)
UQ⊂V UQ⊂V
δv δv
(cid:15)(cid:15) (cid:15)(cid:15)
Ω1(U,t) oo δh C∞(U,T)
U⊂QV⊂W U⊂QV⊂W
δv δv
(cid:15)(cid:15) (cid:15)(cid:15)
. .
. .
. .
10